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Abstract

Using the theory of uniform global attractors for multi-valued semiprocesses, we prove
the existence of attractors for quasilinear parabolic equations related to Caffarelli-Kohn-
Nirenberg inequalities, in which the conditions imposed on the nonlinearity provide
the global existence of weak solutions but not uniqueness, in both autonomous and
non-autonomous cases.

Mathematics Subject Classification 2010: 35B41, 35K65, 35D30.

Keywords:

1. Introduction

The understanding of the asymptotic behavior of dynamical systems is one of the most
important problems of modern mathematical physics. One way to attack the problem for
a dissipative dynamical system is to consider its attractor. The existence of the
attractor has been derived for a large class of PDEs (see e.g., [1,2] and references therein) for both autonomous and non-autonomous equations. However,
these researches may not be applied to a wide class of problems, in which solutions
may not be unique. Good examples of such systems are differential inclusions, variational
inequalities, control infinite dimensional systems and also some partial differential
equations for which solutions may not be known unique as, for example, some certain
semilinear wave equations with high power nonlinearities, the incompressible Navier-Stokes
equation in three space dimension, the Ginzburg-Landau equation, etc. For the qualitative
analysis of the above mentioned systems from the point of view of the theory of dynamical
systems, it is necessary to develop a corresponding theory for multi-valued semigroups.

In the last years, there have been some theories for which one can treat multi-valued
semi-flows and their asymptotic behavior, including the generalized semiflows theory
of Ball [3], theory of trajectory attractors of Chepyzhov and Vishik [4] and theories of multi-valued semiflows and semiprocesses of Melnik and Valero [5-7]. Thanks to these theories, several results concerning attractors in the case of equations
without uniqueness have been obtained recently for differential inclusion [5,6], parabolic equations [8-10], the phase-field equation [11], the wave equation [12], the three-dimensional Navier-Stokes equation [3,13], etc. Although the existence of attractors has been derived for many classes of partial
differential equations without uniqueness, to the best of our knowledge, little seems
to be known for singular/degenerate equations, expecially in the quasilinear case.

Let Ω be a bounded domain in ℝN(N ≥ 2) containing the origin with boundary ∂Ω. In this paper we consider the following quasilinear parabolic equation

(1.1)

where τ ∈ ℝ, uτ ∈ L2(Ω) are given, the nonlinearity f, the external force g, and the numbers p, γ satisfy the following conditions:

(H1) f: ℝ × ℝ → ℝ is a continuous function satisfying

(1.2)

(1.3)

for some q ≥ 2, where C1, C2, k1, k2 are positive constants;

(H2) , where is the set of all translation compact functions in whose definition is given in Definition 1.1 below.

(H3) and .

Let us give some comments about assumptions (H1)-(H3). The nonlinearity f is assumed to have a polynomial growth and to satisfy a standard dissipative condition.
A typical example of functions satisfying conditions (H1) is f (t, u) = |u|q-2u. arctan t, q ≥ 2. We refer the reader to [[1], Chapter 5, Propositions 3.3 and 3.5] for translation compact criterions in . While (H3) is a technical condition ensuring that is embedded compactly into L2(Ω), where is the natural energy space related to problem (1.1), which is defined later in this
section. This is essential for proving the existence of a weak solution to problem
(1.1) using the compactness method.

Problem (1.1), which is related to some Caffarelli-Kohn-Nirenberg inequalities [14], contains some important classes of parabolic equations, such as the semilinear heat
equations (when γ = 0, p = 2), semilinear singular/degenerate parabolic equations (when p = 2), the p-Laplacian equations (when γ = 0, p ≠ 2), etc. The existence and properties of solutions to problem type (1.1) have attracted
interest in recent years [15-19]. However, to the best of our knowledge, little seems to be known for the long-time
behavior of solutions to problem (1.1).

In this article we study the long-time behavior of solutions to problem (1.1) via
the concept of uniform global attractors for multi-valued semiprocesses. Here there
is no restrictions on the growth of the nonlinearity f and the conditions imposed on f provide the global existence of a weak solution to problem (1.1), but not uniqueness.
Thus, when studying the long-time behavior of solutions, in order to handle nonuniqueness
of solutions, we need use the theory of attractors for multi-valued semiprocesses.
Following the general lines of the approach used in [8-10,20] for non-degenerate parabolic equations, we prove the existence of a global compact
attractor in the autonomous case, and of a uniform global compact attractor in the
non-autonomous case. Noting that when the nonlinearity f does not depend on time t, the existence of an attractor for problem (1.1) in the semilinear non-degenerate
case, namely when γ = 0 and p = 2, was studied in [8,9]. Thus, our results extend some known results on the existence and long-time behavior
of solutions of nondegenerate semilinear parabolic equations.

It is worth noticing that under some additional conditions on f, for example, for all t > τ, u ∈ ℝ, or a weaker assumption

one can prove that the weak solution of problem (1.1) is unique. Then the multivalued
semiprocess turns to be a single-valued one and the uniform compact global attractor
is exactly the usual uniform attractor for the family of single-valued semiprocesses
[1].

In the rest of this section, for convenience of the reader, we recall some results
on function spaces related to Caffarelli-Kohn-Nirenberg inequalities and translation
compact functions.

For 1 < p < ∞ and , we define the weighted space

equipped with the norm

It is easy to check that the dual space of is the space , where p' is defined by . Moreover, we define the weighted Sobolev space as the closure of in the norm

(1.4)

As 1 < p < ∞, is reflexive, and the dual space of will be denoted by .

We now state some results which we will use later. The first is the Caffarelli-Kohn-Nirenberg
inequality.

Proposition 1.1. [14]Assume that 1 < p < N. Then there exists a positive constant CN,p,γ,q such that for every ,

From assumption (H3), it is easy to check that there exists a positive number δ such that compactly. Since the embedding is continuous, it is seen that is an evolution triplet.

We now define the following "evolution" spaces which will be useful in what follows.

endowed with the norm

The dual space of is .

Putting

The following proposition, which is easily proved by using similar arguments as in
[[21], Chapter 2], gives some important properties of the operator -Δp,γ.

Proposition 1.2. The operator -Δp,γ maps into its dual . Moreover,

(1) -Δp,γ is hemicontinuous, i.e., for all , the map λ ↦ 〈-Δ p,γ(u + λv), w〉 is continuous from ℝ to ℝ.

(2) -Δp,γ is monotone, i.e., 〈-Δp,γu + Δp,γv,u - v〉 ≥ 0, for all .

Definition 1.1. Assume that ℰ is a reflexive Banach space.

(1) A function is said to be translation bounded if

(2) A function is said to be translation compact if the closure of {φ(⋅ + h)|h ∈ ℝ} is compact in .

Denote by and the sets of all translation bounded functions and of all translation compact functions
in , respectively. It is well-known (see [4]) that .

Let be the closure of the set {g(· + h)|h ∈ ℝ} in . The following results were proved in [[1], Chapter 5, Proposition 3.4].

Lemma 1.3. (1) is compact.

(2) For all ;

(3) The translation group {T(h)}, which is defined by T(h)σ(s) = σ(h + s), s, h ∈ ℝ, is continuous on ;

(4) ;

The rest of the article is organized as follows. In Section 2, we prove the global
existence of a weak solution to problem (1.1) by using the monotonicity and compactness
methods. In Section 3, the existence of global attractors for problem (1.1) is proved
in both the autonomous and non-autonomous cases.

2. Existence of a weak solution

We denote

where p', q' are the conjugate indexes of p, q, respectively.

Definition 2.1. A function u(x, t) is called a weak solution of (1.1) on (τ, T) iff

and

for all test functions φ ∈ V.

It is known (see [[1], Theorem 1.8, p. 33]) that if u ∈ V and , then u ∈ C([τ, T];L2(Ω)). This makes the initial condition in problem (1.1) meaningful.

Step 1: A Galerkin scheme. Consider the approximating solution un(t) in the form

where is a basis of , which is orthonormal in L2(Ω). We get un from solving the problem

Using the Peano theorem in the theory of ODEs, we get the local existence of un.

Step 2: A priori estimates. We have

By assumption (H3), we can choose δ > 0 such that , then and therefore there exists λ > 0 such that

(2.1)

where the last inequality follows from the Young inequality. Using (1.3) and the Cauchy
inequality, we get

Hence

(2.2)

We show that the local solution un can be extended to the interval [τ, ∞). Indeed, from (2.2) we have

By the Gronwall inequality, we obtain

(2.3)

where we have used the facts that and

We now establish some a priori estimates for un. Integrating (2.2) on [τ, T], τ < t ≤ T, and using the fact that , we have

(2.4)

The last inequality implies that

(2.5)

(2.6)

(2.7)

Using hypothesis (1.2), we get

Hence, we can conclude that {f(t, un)} is bounded in and thus,

(2.8)

We have

for all , where we have used the Hölder inequality. Because of the boundedness of {un} in , we infer that {-Δp,γ un} is bounded in .

Step 3: Passing limits. From the above estimates, there exists a subsequence {uμ} ⊂ {un} such that

(2.9)

(2.10)

(2.11)

up to a subsequence.

To prove that η(t) = f(t, u(t)), we argue similarly to [22,23] to deduce that

(2.12)

for all T > τ. In particular, we obtain from (2.5) that

(2.13)

Then, by Theorem 13.3 and Remark 13.1 in [24], we obtain that uμ → u strongly in L2(τ, T; L2(Ω)), up to a subsequence. Hence, we can assume that uμ → u a.e. in Qτ,T. Therefore, f(t, uμ) → f(t, u) a.e. in Qτ,T since f is continuous. By Lemma 1.3 in [[21], Chapter 1], one has

Thus, we have

(2.14)

We now show that ψ = -Δp, γ u. Since -Δp, γ is monotone, we have

Note that {un(T)} is bounded in L2(Ω), so by arguments as in [[21], pp. 159-160], we have that un(T) ⇀ u(T) in L2(Ω). Because

(2.15)

we obtain

(2.16)

where we have used the facts that un(τ) → uτ in . On the other hand, by integrating by parts, from (2.14) we have

and therefore thanks to (2.15) and (2.16) one gets

We now use the hemicontinuity of the operator Δp,γ to show that ψ = -Δp,γ u. Taking v = u - λw, where λ > 0 and , we obtain

hence

(2.17)

leting λ → 0 in (2.17), we conclude that

So ψ = -Δp,γ u. Thus,

We now show that u(τ) = uτ. Choosing some with φ(T) = 0, observe that φ ∈ V, by the Lebesgue dominated theorem, one can check that

Doing the same in the Galerkin approximations yields

Passing to the limit as n → ∞, we have

Therefore, u(τ) = uτ and u is a weak solution of (1.1) on (τ, T).

Finally, it is easy to check that the solution u satisfies the inequality similar to (2.3), and this implies that the solution u exists globally on the interval (τ, +∞).

3. Existence of global attractors

3.1. The autonomous case

Consider the case where f and g do not depend on the time t, and let us recall the definition of multi-valued semiflows.

It is called a strict multi-valued semiflow if , for all w ∈ E, t1, t2 ∈ ℝ+.

We now consider problem (1.1) with τ = 0. By Theorem 2.1, we construct a multi-valued mapping as follows

Lemma 3.1. is a strict multi-valued semiflow in the sense of Definition 3.1.

Proof. Assume that , then ξ = u(t1 + t2), where u(t) is a solution of (1.1). Denoting v (t) = u(t + t2), we see that v(.) is also in the set of solutions of (1.1) with respect to initial condition v(0) = u(t2). Therefore, . It remains to show that . If then ξ = v(t1), where . One can suppose that v(0) = u(t2), where u(0) = u0. Set

Since u and v are two solutions of (1.1), we obtain that w is a solution of (1.1) with w(0) = u(0) = u0. In addition, since ξ = v(t1) = w(t1 + t2), we have .

Definition 3.2. [5]A set is said to be a global attractor of the multi-valued semiflow if the following conditions hold:

• is an attracting, i.e., as t → ∞ for all bounded subsets B ⊂ E,

• is negatively semi-invariant: for arbitrary t ≥ 0,

• If ℬ is an attracting of then ,

where is the Hausdorff semi-distance.

The following theorem gives the sufficient conditions for the existence of a global
attractor for the multi-valued semiflow .

Theorem 3.2. [5,7]Suppose that the strict multi-valued semiflow has the following properties:

(1) is pointwise dissipative, i.e., there exists K > 0 such that for one has ∥u(t)∥E ≤ K if t ≥ t0 (∥u0∥E);

(2) is a closed map for any t ≥ 0, i.e., if ξn → ξ, ηn → η, then ;

(3) is asymptotically upper semicompact, i.e., if B is a bounded set in E such that for
some is bounded, any sequence with tn → ∞ is precompact in E.

Then has a compact global attractor in E. Moreover, is invariant, i.e., for any t ≥ 0.

Lemma 3.3. is a compact mapping for each t* ∈ (0, T].

Proof. This lemma is a direct consequence of Lemma 3.8 in Section 3.2 below.

We now can prove the existence of a global attractor.

Theorem 3.4. Under conditions (H1)-(H3), where f andg are assumed to be independent of time t, the strict multi-valued semiflow
generated by problem (1.1) has an invariant compact global attractor in L2(Ω).

Proof. We will check hypotheses (1)-(3) of Theorem 3.2. First, assume , we have

Noting that

we have

(3.1)

Therefore

Hence one can deduce that is pointwise dissipative.

We now check hypothesis (2) of Theorem 3.2. Assume that in L2(Ω). Then there exists a sequence {un} such that

Using the same arguments as in the proof of Theorem 2.1, we have

• un → u in L2(Q0,T),

• un(t) ⇀ u(t) in L2(Ω) for arbitrary t ∈ [0, T] (and then u(0) = η),

• f(un)⇀ f(u) in ,

• in V,

• -Δp,γ un ⇀ -Δp,γ u in ,

up to a subsequence. Hence, passing to the limit in the equality

we conclude that u(t) is a weak solution of (1.1) with the initial condition u(0) = η. Thus, .

For hypothesis (3), one observes that for n large enough,

where t* > 0 and B* is a bounded set in L2(Ω). Using Lemma 3.3, we conclude that, if , then {ξn} is precompact in L2(Ω).

3.2. The non-autonomous case

Let us recall some definitions and related results. The pair of functions (f(s,⋅),g(⋅,s)) = σ(s) is called a symbol of (1.1). We consider (1.1) with a family of symbols including
the shifted forms σ(s + h) = (f(s + h,⋅), g (⋅, s + h)) and the limits of some sequence {σ(s + hn)}n∈N in an appropriate topological space Σ. The family of such symbols is said to be the
hull of σ in Σ and is denoted by , i.e.,

If the hull is a compact set in Σ, we say that σ is translation compact in Σ.

Denote ℝd = {(t, τ) ∈ ℝ2 | τ ≤ t}. Let X be a complete metric space, and be the set of all nonempty subsets and the set of all nonempty bounded subsets of
the space X, respectively and let Σ be a subspace of Σ.

Denote

Then Z is a Banach space. We say that fn → f in the space C(ℝ; Z) if

(3.2)

for all t ∈ ℝ, r > 0.

Let , and

where the topology in is equipped by the local weak convergence, i.e., gn → g in if

for all t ∈ ℝ, r > 0 and ϕ ∈ L2 (Qt,t+r). We define .

In order to deal with a uniform attractor with respect to the family of symbols, one
usually requires the translation compact property. Let us recall some discussions
on this requirement. It is known that hypothesis (H2) ensures that g is translation compact in (see [4] for details). In addition, the following statement gives a sufficient condition for
the translation compact property in C (ℝ; Z).

Proposition 3.5. [4]The function f ∈ C(ℝ; Z) is translation compact if and only if for all R > 0 one has

From now on, we suppose that f is translation compact. Together with the fact that g is translation compact in , one sees that Σ is a compact set in . Then it follows from [4] that T(h) : Σ → Σ is continuous and T(h)Σ ⊂ Σ for all h ∈ ℝ.

Definition 3.3. [6]The map is called an multi-valued semiprocess (MSP) if

We denote by the set of all global weak solutions (defined for all t ≥ τ) of the problem (1.1) with data (fσ, gσ) instead of (f, g) such that u(τ) = uτ. For each σ = (f, g) ∈ Σ, we consider the family of MSP {Uσ : σ ∈ Σ} defined by

Then the family of multi-valued semiprocesses UΣ has a uniform global compact attractor .

The following is the key point of this subsection.

Lemma 3.8. Let conditions (H1)-(H3) hold and let {un}n∈ℕ is a sequence of weak solutions of (1.1) with respect to the sequence of symbols {σn} ⊂ Σ such that

Then there exists a solution u of (1.1) with respect to the symbol σ such that u(τ) = uT and un(t*) → u(t*) in L2(Ω) for any t* > τ, up to a subsequence.

Proof. Let σn = (fn, gn). Since f satisfies (H1) for all t ∈ ℝ and , one sees that fn also satisfies (H1). On the other hand, noting that {un(τ)} is bounded in L2(Ω) and . Thus, repeating the arguments in the proof of Theorem 2.1, we obtain that

In particular, we have

(3.3)

up to a subsequence. Let in Σ, to show that u is a solution of (1.1) with respect to the symbol σ such that u(τ) = uT, we need to pass to the limits in the following relation

for all v ∈ V. Since in L2(τ,T; L2(Ω)), it remains to prove that in . We first show that in . Indeed,

because in Z and {un} is bounded in Lq(Qτ,T). On the other hand, since is bounded in , by using Lemma 1.3 in [[21], Chapter 1] and the continuity of as in the proof of Theorem 2.1, we can conclude that weakly in . Hence, we have

We now have to show that un(t*) → u(t*) in L2(Ω) for any t* > τ. Taking into account of (3.3), we have to check that .

Putting

It is easy to check that the functions Jn(t), J(t) are continuous and non-increasing on [τ, T]. We first show that

(3.4)

Indeed,

and

as n → ∞ since un → u strongly in L2(Qτ,t) and {gn} is bounded in L2(Qτ,t). In addition,

as n → ∞ since gn ⇀ g in L2(Qτ,t). Then (3.4) is proved due to the fact that un(t) → u(t) in L2(Ω) for a.e. t ∈ [τ, T].

Proof. We know that each symbol σn = (fn, gn) ∈ Σ satisfies the same conditions as in (H1)-(H2). Furthermore, since , we have . Hence if un is a weak solution of (1.1) with respect to the symbol σn, one has

(3.5)

The last inequality ensures the existence of a positive number R0 such that if un(τ) ∈ BR, the ball in L2(Ω) centered at 0 with radius R, then there exists T0 = T0(τ, R) such that

We now define the set . Lemma 3.8 implies that K is compact. Moreover, since is an absorbing set, we have

for all , and t ≥ T0(τ, BR). It follows that any sequence {ξn} such that , is precompact in L2(Ω). It is a consequence of Lemma 3.8 that the map Uσ has compact values for any σ ∈ Σ.

Finally, let us prove that the map (σ, x) ↦ Uσ(t, τ, x) is upper semicontinuous for each fixed t ≥ τ. Suppose that it is not true, that is, there exist , and such that . But Lemma 3.8 implies (up to a subsequence) that , which is a contracdition. Thus, the existence of the uniform global compact attractor
follows then from Theorem 3.7.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Acknowledgements

This work was supported by Vietnam's National Foundation for Science and Technology
Development (NAFOSTED), Project 101.01-2010.05.

The authors would like to thank the reviewers for valuable comments and suggestions.